Question: $ 0.\overline{5} \div 0.\overline{30} = {?} $
Solution: First convert the repeating decimals to fractions. $\begin{align*} 10x &= 5.5555...\\ x &= 0.5555...\end{align*} $ $\begin{align*} 9x &= 5 \\ x &= \dfrac{5}{9}\end{align*} $ $\begin{align*} 100y &= 30.303...\\ y &= 0.303...\end{align*} $ $\begin{align*} 99y &= 30 \\ y &= \dfrac{30}{99}\end{align*} $ So, the problem becomes: $ \dfrac{5}{9} \div \dfrac{30}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{5}{9} \times \dfrac{99}{30} = {?} $ $ \phantom{\dfrac{5}{9} \times \dfrac{30}{99}} = \dfrac{5 \times 99}{9 \times 30} $ $ \phantom{\dfrac{5}{9} \times \dfrac{30}{99}} = \dfrac{5 \times \cancel{99}11} {\cancel{9} \times 30} $ $ \phantom{\dfrac{5}{9} \times \dfrac{30}{99}} = \dfrac{55}{30} $ Simplify: ${= \dfrac{11}{6}}$